Exercise 11.   Consider the complex potential      given implicitly by   .

11 (a).   Show that    determines the ideal fluid flow through an open channel bounded by the rays  

        ,    and    .  

as indicated in Figure 11.109.                     Figure 11.109.

11 (b).   Show that the streamline    of the flow is given by the parametric equations  

       and   ,    for    .

Solution 11.

See text and/or instructor's solution manual.

Answer.   

Solution.   Use the result of Exercise 7 in Section 11.9.  

We showed that      maps the upper half-plane      onto

the upper half-plane      slit along the ray   ,   as shown in Figure 11.81.  

Observe that this function will also map a flow across the entire z-plane  (minus the x-axis) onto the region show in Figure 11.109.

Set  ,  ,  and  ,  ,  respectively, and let  .    

The exterior angles are  ,  and the formula for the derivative    is  

               
Integrate and get

            

            

            
            

The graph for the streamlines can be produced by using the formula      and making the substitution    for  .  

Hence, we obtain   .  

Then the image of the line      will be  

                      

Thus the parametric equations for the streamlines will be  

                    ,   and  

                    ,   for  .  

We are really done.   

        We can let Mathematica draw a graph of the streamlines.

                    The streamlines  .  

We are really really done.   

        We can use Mathematica to graph the conformal mapping   .  

                      The conformal mapping   .  

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell